Theorem lcfrlem17

Description: Lemma for lcfr . Condition needed more than once. (Contributed by NM, 11-Mar-2015)

	Ref	Expression
Hypotheses	lcfrlem17.h	|- H = ( LHyp ` K )
	lcfrlem17.o	|- ._|_ = ( ( ocH ` K ) ` W )
	lcfrlem17.u	|- U = ( ( DVecH ` K ) ` W )
	lcfrlem17.v	|- V = ( Base ` U )
	lcfrlem17.p	|- .+ = ( +g ` U )
	lcfrlem17.z	|- .0. = ( 0g ` U )
	lcfrlem17.n	|- N = ( LSpan ` U )
	lcfrlem17.a	|- A = ( LSAtoms ` U )
	lcfrlem17.k	|- ( ph -> ( K e. HL /\ W e. H ) )
	lcfrlem17.x	|- ( ph -> X e. ( V \ { .0. } ) )
	lcfrlem17.y	|- ( ph -> Y e. ( V \ { .0. } ) )
	lcfrlem17.ne	|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
Assertion	lcfrlem17	|- ( ph -> ( X .+ Y ) e. ( V \ { .0. } ) )

Proof

Step	Hyp	Ref	Expression
1		lcfrlem17.h	|- H = ( LHyp ` K )
2		lcfrlem17.o	|- ._|_ = ( ( ocH ` K ) ` W )
3		lcfrlem17.u	|- U = ( ( DVecH ` K ) ` W )
4		lcfrlem17.v	|- V = ( Base ` U )
5		lcfrlem17.p	|- .+ = ( +g ` U )
6		lcfrlem17.z	|- .0. = ( 0g ` U )
7		lcfrlem17.n	|- N = ( LSpan ` U )
8		lcfrlem17.a	|- A = ( LSAtoms ` U )
9		lcfrlem17.k	|- ( ph -> ( K e. HL /\ W e. H ) )
10		lcfrlem17.x	|- ( ph -> X e. ( V \ { .0. } ) )
11		lcfrlem17.y	|- ( ph -> Y e. ( V \ { .0. } ) )
12		lcfrlem17.ne	|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
13	1 3 9	dvhlmod	|- ( ph -> U e. LMod )
14	10	eldifad	|- ( ph -> X e. V )
15	11	eldifad	|- ( ph -> Y e. V )
16	4 5	lmodvacl	|- ( ( U e. LMod /\ X e. V /\ Y e. V ) -> ( X .+ Y ) e. V )
17	13 14 15 16	syl3anc	|- ( ph -> ( X .+ Y ) e. V )
18	4 5 6 7 13 14 15 12	lmodindp1	|- ( ph -> ( X .+ Y ) =/= .0. )
19		eldifsn	|- ( ( X .+ Y ) e. ( V \ { .0. } ) <-> ( ( X .+ Y ) e. V /\ ( X .+ Y ) =/= .0. ) )
20	17 18 19	sylanbrc	|- ( ph -> ( X .+ Y ) e. ( V \ { .0. } ) )